Regression Test of Group Equality via URNMODEL
----------------------------------------------

AUTHOR:   Richard Goldstein, Qualitas
SUPPORT:  Written communication only, EMAIL goldst@@harvarda.bitnet
	  or 37 Kirkwood Road, Brighton MA 02135.

                 ^urnmodel^ varlist [^in^ range] [^if^ exp]

^urnmodel^ uses an approximation to a randomization test on the residuals of a
standard linear regression to test the equality of two groups (e.g., males and
females)--^DO NOT^ include a dummy variable for these groups in your regression.
The ^first^ variable in varlist should be the group variable to be tested--the 
code expects this variable to be coded as a 0-1 dummy variable.  The ^second^ 
variable in varlist is the dependent variable for the regression, and the 
remaining variables are the right-hand-side variables for the regression.

This model is discussed in some detail in Levin, B and Robbins, H (1983), "Urn
Models for Regression Analysis, with Applications to Employment Discrimination
Studies", ^Law and Contemporary Problems^, 46, pp. 247-67, and more briefly in
Finkelstein, MO and Levin, B (1990), ^Statistics for Lawyers^, New York:
Springer-Verlag, esp. at pp. 399-402.  A strata-oriented extension of this
model is described on pp. 253-5 of Levin and Robbins.

Note that this model should be "less significant" than a model that (1)
includes a sex coefficient, or (2) includes sex by variable interactions,
since "To the extent that sex is a factor in determining salary, and is
correlated with the productivity [sic] factors, its effect is assigned to
those productivity [sic] factors."  (F&L, p. 402) Among other things, this
means that this procedure will generally have lower power than either of the
above two procedures.  Both citations show the algebraic relationship between
the z-test calculated here for the urn model and the t-test resulting from a
regression with a dummy variable for sex.

If you have available a randomization t-test procedure, that will be more
accurate than the approximation used here:  write out the residuals with the
grouping variable to the other package and use the exact randomization test
there.  If they only have an approximate randomization test, you should
probably use both ^urnmodel^ and the approximate test.  The approximation used
here is based on the normal distribution.

Example:
--------

------------------------------------------------------------------------------
 . ^use auto^
 (1978 Automobile Data)

 . ^urnmodel foreign mpg weight weightsq^
 (obs=74)

   Source |     SS         df       MS           Number of obs =      74
 ---------+------------------------------        F(  2,    71) =   72.80
    Model |  1642.52197     2  821.260986        Prob > F      =  0.0000
 Residual |  800.937487    71  11.2808097        R-square      =  0.6722
 ---------+------------------------------        Adj R-square  =  0.6630
    Total |  2443.45946    73  33.4720474        Root MSE      =  3.3587

 Variable |  Coefficient    Std. Error       t    Prob > |t|        Mean
 ---------+--------------------------------------------------------------
      mpg |                                                      21.2973
 ---------+--------------------------------------------------------------
   weight |    -.0141581      .0038835    -3.646     0.001      3019.459
 weightsq |     1.32e-06      6.26e-07     2.116     0.038       9713003
    _cons |     51.18308      5.767884     8.874     0.000             1
 ---------+--------------------------------------------------------------
 
 Summary of Residuals for Group=0

 Variable |   Obs        Mean   Std. Dev.       Min        Max
 ---------+---------------------------------------------------
     _res |    52    .4069727   2.198623  -2.385209   8.896997

 Summary of Residuals for Group=1

 Variable |   Obs        Mean   Std. Dev.       Min        Max
 ---------+---------------------------------------------------
     _res |    22   -.9619354   5.002079  -6.754405   13.18774

The test statistic is z= 1.6249259 and its p-value is 0.0521
------------------------------------------------------------------------------

Let's compare this to the same model with a dummy variable:


------------------------------------------------------------------------------
 . ^reg mpg weight weightsq foreign^
 (obs=74)

   Source |     SS         df       MS           Number of obs =      74
 ---------+------------------------------        F(  3,    70) =   52.25
    Model |  1689.15372     3   563.05124        Prob > F      =  0.0000
 Residual |   754.30574    70  10.7757963        R-square      =  0.6913
 ---------+------------------------------        Adj R-square  =  0.6781
    Total |  2443.45946    73  33.4720474        Root MSE      =  3.2827

 Variable |  Coefficient    Std. Error       t    Prob > |t|        Mean
 ---------+--------------------------------------------------------------
      mpg |                                                      21.2973
 ---------+--------------------------------------------------------------
   weight |    -.0165729      .0039692    -4.175     0.000      3019.459
 weightsq |     1.59e-06      6.25e-07     2.546     0.013       9713003
  foreign |      -2.2035      1.059246    -2.080     0.041      .2972973
    _cons |     56.53884      6.197383     9.123     0.000             1
 ---------+--------------------------------------------------------------
 
------------------------------------------------------------------------------
Note that, as expected, the p-value for foreign is more significant when used
as a dummy variable than with the approximate randomization test.  However,
even though the power is lower, it is sometimes desirable to use the
randomization procedure; see the above article for more.  This is ^not^ a
randomization regression, though there are such models in the literature.